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VCE Foundation Mathematics (Year 12) · Number and Arithmetic · 10m read

Integers, Fractions and Decimals

Ever notice how the same amount can be written different ways—$5, 5.00, or 5/1? Understanding integers, fractions, and decimals means you can switch between them smoothly and tackle real problems like splitting a pizza bill, measuring ingredients, or working out discounts. This skill unlocks confidence across VCE maths and everyday life.

Learning objective: Convert confidently between integers, fractions, decimals and percentages and apply them to everyday money and measurement problems.

By the end of this lesson you can:

  • I can convert a fraction to a decimal and to a percentage without a calculator for common values.
  • I can add and subtract fractions by finding a common denominator.
  • I can operate with negative numbers using a number line.

What are integers, fractions and decimals

Numbers come in different forms and each tells you something different about the world. Integers are whole numbers — positive, negative or zero. Fractions show parts of a whole using a top number and a bottom number. Decimals are fractions written in base ten. Percentages are fractions out of 100.

In the VCAA Foundation Maths exam these forms appear constantly in real-world money and measurement contexts. The speed comes from being fluent in converting between them — knowing instinctively that 3/8 = 0.375 = 37.5% saves time on every multi-step question.

Same pizza, three languages. You order a pizza cut into 8 slices and eat 3. That's the fraction 3/8. Divide 3 by 8 on your calculator: 0.375, the decimal. Multiply by 100: 37.5, the percentage. Same amount of pizza — three ways of describing it. The whole pizza before you touched it was the integer 1.

Conversion shortcuts

  • Fraction → decimal: Divide top by bottom. 3 ÷ 8 = 0.375.
  • Decimal → percentage: Multiply by 100. 0.375 × 100 = 37.5%.
  • Adding fractions: Make the bottoms match first. 1/4 + 1/3 = 3/12 + 4/12 = 7/12.
  • Negative numbers: On the number line they go left. −5 is less than −2.

Quick reference

| Skill | Example |
|---|---|
| Integer add | −5 + 8 = 3 |
| Fraction → decimal | 3/4 = 0.75 |
| Add fractions | ½ + ⅓ = 5/6 |
| Decimal × 10 | 3.47 × 10 = 34.7 |
| Rounding | 2.678 → 2.68 (2 d.p.) |

Worked example (Medium)

Sarah earns $240 per week. She spends 3/8 on rent and 0.2 on food. How much does she save each week?

  1. Convert 3/8 to a decimal. Divide top by bottom. 3 ÷ 8 = 0.375
  2. Rent amount. Multiply weekly earnings by the rent fraction. $240 × 0.375 = $90
  3. Food amount. Multiply weekly earnings by the food decimal. $240 × 0.2 = $48
  4. Savings. Subtract rent and food from total earnings. $240 − $90 − $48 = $102

Answer: Sarah saves $102 per week.

Practice questions

  1. A footy team won 9 out of 16 games. Write their win rate as a fraction, decimal and percentage.

    1. 9/16 = 0.5625 = 56.25%
    2. 9/16 = 0.5625 = 56%
    3. 9/16 = 0.625 = 62.5%
    4. 9/16 = 0.5 = 50%

    (2 marks)

  2. The temperature in Ballarat was −4°C at 6am. It rose 11°C by noon then dropped 7°C by 6pm. What was the temperature at 6pm?

    1. 0°C
    2. −1°C
    3. 7°C
    4. 14°C

    (2 marks)

  3. A canteen buys 6.5 kg of chicken at $8.40/kg. They use 3/4 of the chicken on Monday and 0.15 of the original amount on Tuesday. How many kg are left?

    1. 0.65 kg
    2. 1.625 kg
    3. 0.975 kg
    4. 5.85 kg

    (3 marks)

  4. A construction site starts with 12.5 tonnes of sand. They use 2/5 on Wednesday and 3.2 tonnes on Thursday. On Friday a delivery increases the remaining amount by 50%. How much sand is on site after Friday?

    (3 marks)

  5. Express 7/20 as both a decimal and a percentage.

    1. 0.35 = 35%
    2. 0.7 = 70%
    3. 0.20 = 20%
    4. 0.45 = 45%

    (2 marks)

  6. Order from smallest to largest: 0.7, 3/4, 70%, 0.65.

    1. 0.65, 0.7, 70%, 3/4
    2. 3/4, 70%, 0.7, 0.65
    3. 0.7, 70%, 0.65, 3/4
    4. 0.65, 3/4, 0.7, 70%

    (2 marks)

Worked example (Hard)

A 25 m roll of rope is cut so that 3/10 is used for fencing, 0.45 of the original is used for a clothesline, and the rest is divided equally between 4 garden beds. How many metres does each garden bed receive?

  1. Convert 3/10. Divide top by bottom. 3 ÷ 10 = 0.30
  2. Fencing length. Multiply original length by the fencing decimal. 25 × 0.30 = 7.5 m
  3. Clothesline length. Multiply original length by the clothesline decimal. 25 × 0.45 = 11.25 m
  4. Remaining rope. Subtract both uses from the original. 25 − 7.5 − 11.25 = 6.25 m
  5. Per garden bed. Divide remainder by 4 beds. 6.25 ÷ 4 = 1.5625 m

Answer: Each garden bed receives 1.5625 m of rope (about 1.56 m).

Common mistakes

  • Treating 3/8 as 3.8 or 0.38 A fraction line means division. 3/8 is 3 ÷ 8 = 0.375, not the digits "3" and "8" run together. Fix: Always divide the top number by the bottom number on the calculator before substituting.
  • Adding fractions by adding tops and bottoms 1/4 + 1/3 ≠ 2/7. You must rewrite both fractions over a common denominator first. Fix: Use the LCM of the denominators: 1/4 + 1/3 = 3/12 + 4/12 = 7/12.
  • Rounding negative numbers the wrong way −5 is *less than* −2, but on first glance students treat the bigger digit as the bigger number. Fix: Plot both on a number line — left = smaller, right = larger.

Going deeper — terminating vs recurring decimals

A fraction in lowest terms has a TERMINATING decimal expansion (e.g. 0.375) only when its denominator's prime factors are 2s and 5s — otherwise the expansion repeats forever (e.g. 1/3 = 0.333…). This is why 3/8 stops dead but 1/7 cycles through 6 digits before repeating.

To compare fractions quickly without a calculator: cross-multiply. For 3/8 vs 5/13, compare 3 × 13 = 39 against 5 × 8 = 40. Bigger cross-product wins, so 5/13 > 3/8. This trick saves time on multiple-choice number-form questions.

Summary

  • Three forms of the same number: fraction (top ÷ bottom), decimal, percentage (× 100).
  • Add fractions by finding a common denominator before combining the tops.
  • Negative numbers go LEFT on the number line — −5 < −2.
  • In money or measurement, convert to decimals first, then calculate, then round only at the end.

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